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Omegaile comments on Solving the two envelopes problem - Less Wrong
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There is another very cool puzzle that can be considered a followup which is:
There are two envelopes in which I, the host of the game, put two different natural numbers, chosen by any distribution I like, that you don't have access. The two envelopes are indistinguishable. You pick one of them (and since they are indistinguishable, this can be considered a fair coin flip). After that you open the envelope and see the number. You have a chance to switch your number for the hidden number. Then, this number is revealed and if you choose the greater you win, let's say a dollar, otherwise you pay a dollar.
Now, before everything I said happens, you must devise a strategy that guarantees that you have a greater than 1/2 chance of winning.
Some notes:
1- the problem may be extended for rational, or any set of constructive numbers. But if you want to think only in probabilities this is irrelevant, just an over formalism.
2- This may seem uncorrelated to the two envelopes puzzle at first, but it isn't.
3- I saw this problem first on EDITthis post on xkcd blag. Thanks for Vaniver for pointing out.
I believe you're thinking of this blag post.
Yes, that's it! Thanks.
Isn't there an additional requirement that there is a minimum element in the set?
No, you can think on the rationals, for example.
Well natural numbers and simple greater than satisfying makes it easy. "If one THEN swap ELSE keep."
Maybe I didn't express myself well, but this strategy should work regardless of the distribution I choose. For example, if I choose a distribution in which 1 has probability 0, than your strategy yield 1/2 chance.
If that kind of selection of distributions is possible then there is no free lunch to be found.
For any strategy of envelope switching a hostile distribution selector who knows your strategy in advance can trivially select distributions to thwart it.
Have you looked at the "solution"? There really isn't a counter-strategy that reduces it to 1/2 chance, although there are strategies moving it arbitrarily close to 1/2.